3.32 \(\int \frac{\log (e (f (a+b x)^p (c+d x)^q)^r)}{(g+h x)^3} \, dx\)

Optimal. Leaf size=202 \[ \frac{b^2 p r \log (a+b x)}{2 h (b g-a h)^2}-\frac{b^2 p r \log (g+h x)}{2 h (b g-a h)^2}-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac{b p r}{2 h (g+h x) (b g-a h)}+\frac{d^2 q r \log (c+d x)}{2 h (d g-c h)^2}-\frac{d^2 q r \log (g+h x)}{2 h (d g-c h)^2}+\frac{d q r}{2 h (g+h x) (d g-c h)} \]

[Out]

(b*p*r)/(2*h*(b*g - a*h)*(g + h*x)) + (d*q*r)/(2*h*(d*g - c*h)*(g + h*x)) + (b^2*p*r*Log[a + b*x])/(2*h*(b*g -
a*h)^2) + (d^2*q*r*Log[c + d*x])/(2*h*(d*g - c*h)^2) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(2*h*(g + h*x)^2)
- (b^2*p*r*Log[g + h*x])/(2*h*(b*g - a*h)^2) - (d^2*q*r*Log[g + h*x])/(2*h*(d*g - c*h)^2)

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Rubi [A]  time = 0.111538, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2495, 44} \[ \frac{b^2 p r \log (a+b x)}{2 h (b g-a h)^2}-\frac{b^2 p r \log (g+h x)}{2 h (b g-a h)^2}-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac{b p r}{2 h (g+h x) (b g-a h)}+\frac{d^2 q r \log (c+d x)}{2 h (d g-c h)^2}-\frac{d^2 q r \log (g+h x)}{2 h (d g-c h)^2}+\frac{d q r}{2 h (g+h x) (d g-c h)} \]

Antiderivative was successfully verified.

[In]

Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(g + h*x)^3,x]

[Out]

(b*p*r)/(2*h*(b*g - a*h)*(g + h*x)) + (d*q*r)/(2*h*(d*g - c*h)*(g + h*x)) + (b^2*p*r*Log[a + b*x])/(2*h*(b*g -
a*h)^2) + (d^2*q*r*Log[c + d*x])/(2*h*(d*g - c*h)^2) - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(2*h*(g + h*x)^2)
- (b^2*p*r*Log[g + h*x])/(2*h*(b*g - a*h)^2) - (d^2*q*r*Log[g + h*x])/(2*h*(d*g - c*h)^2)

Rule 2495

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((g_.) + (h_.)*(x_))^(m_.),
x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(h*(m + 1)), x] + (-Dist[(b*p*r)/(
h*(m + 1)), Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Dist[(d*q*r)/(h*(m + 1)), Int[(g + h*x)^(m + 1)/(c + d*x
), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(g+h x)^3} \, dx &=-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac{(b p r) \int \frac{1}{(a+b x) (g+h x)^2} \, dx}{2 h}+\frac{(d q r) \int \frac{1}{(c+d x) (g+h x)^2} \, dx}{2 h}\\ &=-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}+\frac{(b p r) \int \left (\frac{b^2}{(b g-a h)^2 (a+b x)}-\frac{h}{(b g-a h) (g+h x)^2}-\frac{b h}{(b g-a h)^2 (g+h x)}\right ) \, dx}{2 h}+\frac{(d q r) \int \left (\frac{d^2}{(d g-c h)^2 (c+d x)}-\frac{h}{(d g-c h) (g+h x)^2}-\frac{d h}{(d g-c h)^2 (g+h x)}\right ) \, dx}{2 h}\\ &=\frac{b p r}{2 h (b g-a h) (g+h x)}+\frac{d q r}{2 h (d g-c h) (g+h x)}+\frac{b^2 p r \log (a+b x)}{2 h (b g-a h)^2}+\frac{d^2 q r \log (c+d x)}{2 h (d g-c h)^2}-\frac{\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2}-\frac{b^2 p r \log (g+h x)}{2 h (b g-a h)^2}-\frac{d^2 q r \log (g+h x)}{2 h (d g-c h)^2}\\ \end{align*}

Mathematica [A]  time = 0.549866, size = 206, normalized size = 1.02 \[ \frac{\frac{r (g+h x) \left ((b c-a d) (b g-a h) (d g-c h) (b d g (p+q)-h (a d q+b c p))-(g+h x) \left (d^2 q (a d-b c) (b g-a h)^2 (\log (c+d x)-\log (g+h x))-b^2 p (b c-a d) (d g-c h)^2 (\log (a+b x)-\log (g+h x))\right )\right )}{(b c-a d) (b g-a h)^2 (d g-c h)^2}-\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{2 h (g+h x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(g + h*x)^3,x]

[Out]

(-Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] + (r*(g + h*x)*((b*c - a*d)*(b*g - a*h)*(d*g - c*h)*(b*d*g*(p + q) - h*
(b*c*p + a*d*q)) - (g + h*x)*(-(b^2*(b*c - a*d)*(d*g - c*h)^2*p*(Log[a + b*x] - Log[g + h*x])) + d^2*(-(b*c) +
a*d)*(b*g - a*h)^2*q*(Log[c + d*x] - Log[g + h*x]))))/((b*c - a*d)*(b*g - a*h)^2*(d*g - c*h)^2))/(2*h*(g + h*
x)^2)

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Maple [F]  time = 0.496, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( e \left ( f \left ( bx+a \right ) ^{p} \left ( dx+c \right ) ^{q} \right ) ^{r} \right ) }{ \left ( hx+g \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^3,x)

[Out]

int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^3,x)

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Maxima [A]  time = 1.25929, size = 313, normalized size = 1.55 \begin{align*} \frac{{\left (b f p{\left (\frac{b \log \left (b x + a\right )}{b^{2} g^{2} - 2 \, a b g h + a^{2} h^{2}} - \frac{b \log \left (h x + g\right )}{b^{2} g^{2} - 2 \, a b g h + a^{2} h^{2}} + \frac{1}{b g^{2} - a g h +{\left (b g h - a h^{2}\right )} x}\right )} + d f q{\left (\frac{d \log \left (d x + c\right )}{d^{2} g^{2} - 2 \, c d g h + c^{2} h^{2}} - \frac{d \log \left (h x + g\right )}{d^{2} g^{2} - 2 \, c d g h + c^{2} h^{2}} + \frac{1}{d g^{2} - c g h +{\left (d g h - c h^{2}\right )} x}\right )}\right )} r}{2 \, f h} - \frac{\log \left (\left ({\left (b x + a\right )}^{p}{\left (d x + c\right )}^{q} f\right )^{r} e\right )}{2 \,{\left (h x + g\right )}^{2} h} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^3,x, algorithm="maxima")

[Out]

1/2*(b*f*p*(b*log(b*x + a)/(b^2*g^2 - 2*a*b*g*h + a^2*h^2) - b*log(h*x + g)/(b^2*g^2 - 2*a*b*g*h + a^2*h^2) +
1/(b*g^2 - a*g*h + (b*g*h - a*h^2)*x)) + d*f*q*(d*log(d*x + c)/(d^2*g^2 - 2*c*d*g*h + c^2*h^2) - d*log(h*x + g
)/(d^2*g^2 - 2*c*d*g*h + c^2*h^2) + 1/(d*g^2 - c*g*h + (d*g*h - c*h^2)*x)))*r/(f*h) - 1/2*log(((b*x + a)^p*(d*
x + c)^q*f)^r*e)/((h*x + g)^2*h)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)/(h*x+g)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.57893, size = 1413, normalized size = 7. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(h*x+g)^3,x, algorithm="giac")

[Out]

-1/2*p*r*log(b*x + a)/(h^3*x^2 + 2*g*h^2*x + g^2*h) - 1/2*q*r*log(d*x + c)/(h^3*x^2 + 2*g*h^2*x + g^2*h) - 1/2
*(b^2*d^2*g^2*p*r - 2*b^2*c*d*g*h*p*r + b^2*c^2*h^2*p*r + b^2*d^2*g^2*q*r - 2*a*b*d^2*g*h*q*r + a^2*d^2*h^2*q*
r)*log(h*x + g)/(b^2*d^2*g^4*h - 2*b^2*c*d*g^3*h^2 - 2*a*b*d^2*g^3*h^2 + b^2*c^2*g^2*h^3 + 4*a*b*c*d*g^2*h^3 +
a^2*d^2*g^2*h^3 - 2*a*b*c^2*g*h^4 - 2*a^2*c*d*g*h^4 + a^2*c^2*h^5) + 1/4*(b^2*d^2*g^2*p*r - 2*b^2*c*d*g*h*p*r
+ b^2*c^2*h^2*p*r + b^2*d^2*g^2*q*r - 2*a*b*d^2*g*h*q*r + a^2*d^2*h^2*q*r)*log(abs(b*d*x^2 + b*c*x + a*d*x +
a*c))/(b^2*d^2*g^4*h - 2*b^2*c*d*g^3*h^2 - 2*a*b*d^2*g^3*h^2 + b^2*c^2*g^2*h^3 + 4*a*b*c*d*g^2*h^3 + a^2*d^2*g
^2*h^3 - 2*a*b*c^2*g*h^4 - 2*a^2*c*d*g*h^4 + a^2*c^2*h^5) + 1/2*(b*d*g*h*p*r*x - b*c*h^2*p*r*x + b*d*g*h*q*r*x
- a*d*h^2*q*r*x + b*d*g^2*p*r - b*c*g*h*p*r + b*d*g^2*q*r - a*d*g*h*q*r - b*d*g^2*r*log(f) + b*c*g*h*r*log(f)
+ a*d*g*h*r*log(f) - a*c*h^2*r*log(f) - b*d*g^2 + b*c*g*h + a*d*g*h - a*c*h^2)/(b*d*g^2*h^3*x^2 - b*c*g*h^4*x
^2 - a*d*g*h^4*x^2 + a*c*h^5*x^2 + 2*b*d*g^3*h^2*x - 2*b*c*g^2*h^3*x - 2*a*d*g^2*h^3*x + 2*a*c*g*h^4*x + b*d*g
^4*h - b*c*g^3*h^2 - a*d*g^3*h^2 + a*c*g^2*h^3) + 1/4*(b^3*c*d^2*g^2*p*r - a*b^2*d^3*g^2*p*r - 2*b^3*c^2*d*g*h
*p*r + 2*a*b^2*c*d^2*g*h*p*r + b^3*c^3*h^2*p*r - a*b^2*c^2*d*h^2*p*r - b^3*c*d^2*g^2*q*r + a*b^2*d^3*g^2*q*r +
2*a*b^2*c*d^2*g*h*q*r - 2*a^2*b*d^3*g*h*q*r - a^2*b*c*d^2*h^2*q*r + a^3*d^3*h^2*q*r)*log(abs((2*b*d*x + b*c +
a*d - abs(-b*c + a*d))/(2*b*d*x + b*c + a*d + abs(-b*c + a*d))))/((b^2*d^2*g^4*h - 2*b^2*c*d*g^3*h^2 - 2*a*b*
d^2*g^3*h^2 + b^2*c^2*g^2*h^3 + 4*a*b*c*d*g^2*h^3 + a^2*d^2*g^2*h^3 - 2*a*b*c^2*g*h^4 - 2*a^2*c*d*g*h^4 + a^2*
c^2*h^5)*abs(-b*c + a*d))